Hyperminrank: Unified Multi-Sender Coding Metric

• Hyperminrank is a scalar linear invariant defined via the sum of matrix ranks over composite adjacency matrices that fit a 4-regular side-information hypergraph.
• It unifies traditional single-sender minrank formulations with multi-sender index coding by capturing sender interactions, message placements, and receiver side-information in a hypergraph model.
• The framework offers tight achievability-converse duality and practical applications in cache-aided communication, distributed storage, coded computation, and edge networking.

Hyperminrank is the scalar linear optimization invariant defined in “Hyper-Minrank: A Unified Hypergraph Characterization of Multi-Sender Index Coding” as the exact broadcast length for linear multi-sender index coding (MSIC) problems via a 4-regular side-information hypergraph construction. It generalizes minrank formulations of traditional single-sender index coding to the multi-sender regime by encoding the interactions among senders, message placements, and receiver side-information as hypergraph-theoretic objects with composite adjacency representations. Hyperminrank enables tight achievable and converse characterizations, algorithmic evaluation, and unifies the analysis of multi-sender coding in cache-aided communication, distributed storage, coded computation, edge networking, and related domains (Khalesi et al., 15 Dec 2025).

1. Multi-Sender Index Coding and Hypergraph Model

The standard MSIC setup consists of $N$ senders, each storing a subset $M_n \subseteq \{1, \ldots, K\}$ of $K$ binary messages $x_1, \ldots, x_K$, and $K$ receivers, where receiver $k$ requests $x_k$ and has a side-information set $R(k) \subset \{1, \ldots, K\} \setminus \{k\}$.

The core abstraction is a directed 4-regular side-information hypergraph $\mathcal{G} = (\mathcal{V},\mathcal{E})$, where vertices are pairs $(k, S)$ for $k \in [K]$ and $S \subset [N]$ with $|S| \leq 2$. Each hyperedge, corresponding to a tuple $(k, k′, n, n′)$ with $\{n, n′\} = S$, represents:

• Demand edges $\mathcal{E}_d$: $(k, k, n, n)$, indicating $x_k$ stored at sender $n$.
• Cached-data edges $\mathcal{E}_s$: $(k, k', n, n)$, $x_{k′}$ known at receiver $k$, stored at sender $n$.
• Coupled edges $\mathcal{E}_c$: $(k, k′, n, n′)$, $x_{k′}$ held at both $n,n′$ but not in $R(k)$.

A composite adjacency matrix $A_{\mathcal{G}} = [A_1 A_2 \ldots A_N]$, where each $A_n \in \mathbb{F}_2^{K \times K}$ indicates which sender/message combinations contribute to each receiver, encodes the valid transmission options.

2. Definition and Properties of Hyperminrank

Hyperminrank is defined via the sum of matrix ranks over the composite adjacency fitting valid hypergraph substructures:

Valid sub-hypergraph: $\mathcal{G}^\prime \subseteq \mathcal{G}$ is valid if every receiver $k$ is incident to an odd number of demand edges, ensuring decodability for $x_k$ under scalar linear coding.

Fitting: $A=[A_1 \ldots A_N]$ fits $\mathcal{G}$ if it is a composite adjacency for a valid $\mathcal{G}'$.

The hyperminrank is

$\operatorname{hyper\text{-}minrank}(\mathcal{G}) = \min_{\text{$A$fits$\mathcal{G}$}} \sum_{n=1}^N \operatorname{rank}(A_n).$

This is the minimal total scalar linear transmission length over all senders, generalizing the single-sender minrank to arbitrary message replication and cross-sender dependencies.

3. Achievability-Converse Duality and Main Theorem

The pivotal result is the tight achievability-converse equivalence:

Main Theorem: For any side-information hypergraph $\mathcal{G}$, the optimal scalar linear broadcast length $$\ell_\mathrm{lin}^*(\mathcal{G}) = \operatorname{hyper\text{-}minrank}(\mathcal{G})$$.

• Achievability: Given a fitting $A$ minimizing $\sum \operatorname{rank}(A_n)$, each sender transmits a basis for its row space; each receiver can decode its demanded message using the parity structure imposed by the valid fitting and cancel known side-information.
• Converse: Any scalar linear code naturally determines such a fitting. The code construction implies the existence of an adjacency pattern, whose sum-rank must be less than or equal to the number of transmissions. Optimizing over all codes yields the hyperminrank lower bound.

This equivalence unambiguously links index code length to a hypergraph invariant, subsuming previous minrank and fitting-matrix approaches for the multi-sender case.

4. Hypergraph-Theoretic Bounds—Clique-Cover and Complement

Hyperminrank facilitates direct analogues of classical graph bounds (Haemers-type) for the multi-sender scenario:

• Clique-cover upper bound: If $[K]$ can be covered by $m$ valid hypergraphic cliques (each implementable by a sender), then $$\operatorname{hyper\text{-}minrank}(\mathcal{G}) \leq m$$. A hypergraphic clique $C$ is valid if each sender-vertex in $S(C)$ has odd degree in the single-sender projection, so each associated $A_n$ has rank one.
• Complement-hypergraph lower bound: Let $\overline{\mathcal{G}}$ be the complement, defined by taking single-sender non-edges from $\mathcal{G}$ as edges, forbidding singleton projections of coupled-edges. The maximum receiver-clique size contained entirely in a single-sender projection provides a lower bound: $$|\mathcal{C}| \leq \operatorname{hyper\text{-}minrank}(\mathcal{G})$$.

This duality sharpens both constructive and impossibility results for broadcast length optimization.

5. Algorithmic Evaluation and Complexity

An exhaustive algorithm computes $\operatorname{hyper\text{-}minrank}(\mathcal{G})$ by enumerating all valid sub-hypergraphs subject to the receiver-local parity and structural constraints. For each receiver $k$:

• Choose an odd subset of demand edges, arbitrary subsets of cached-data edges, and for each $k'\neq k$, an even-sized subset of coupled-sender pairs.
• For each such selection, build $A_n$, compute ranks, and track the minimum sum.

The search space exponent is

$$O\left( \sum_k \left( |R(k)| + \sum_{m\notin R(k)} (|M(m)|-1) \right) \right),$$

times a polynomial factor for rank computation. This is asymptotically faster than the LT–CMAR heuristic (which grows quadratically in per-sender message count) when message replication and side-information sets are bounded. It also improves over Kim–No's exhaustive fitting-matrix search except in cases with no redundant storage (Khalesi et al., 15 Dec 2025).

6. Illustrative Example

Given $K=N=3$ with $M_1 = \{1,2\}$, $M_2 = \{2,3\}$, $M_3 = \{1,3\}$, and side-information $R(1)=\{2\},\, R(2)=\{3\},\, R(3)=\{3\}$:

• $\mathcal{G}$ has three demand edges per sender, plus cached and coupled edges.
• A valid fitting $A'$ realizes $\sum \operatorname{rank}(A_n) = 3$ by setting:
  • $A'_1$ row 1, cols [1,2] = [1,1];
  • $A'_2$ row 2, cols [2,3] = [1,1];
  • $A'_3$ row 3, cols [1,3] = [1,1].
• Corresponding transmissions: $S_1: x_1 + x_2$, $S_2: x_2 + x_3$, $S_3: x_1 + x_3$, enabling decoding for all receivers in 3 transmissions.
• No lower sum-rank fitting exists, so $$\operatorname{hyper\text{-}minrank}(\mathcal{G}) = 3$$.
• Clique-cover upper bound equals 3; complement-clique lower bound yields 2.

Aspect	Example Value	Consequence
Demand edge count	3 per sender	Governs decoding structure
Clique-cover upper bd	3	Achieved by 3 single-sender cliques
Complement clique bd	2	Tight lower bound

7. Applications Across Communication and Storage Systems

The hyperminrank framework precisely characterizes the optimal scalar linear resource requirements for:

• Multi-sender cache-aided communication: Coupled edges model cross-cache interference and cooperative coded delivery.
• MapReduce and coded distributed computation: Hyperminrank dualizes to the minimum linear communication load given partial data sharding.
• Distributed storage and repair: Hypergraph structure mirrors fragment placements; minrank yields minimum repair bandwidth.
• Edge/satellite networks: Guides optimal multicast scheduling across multiple gateways with partial content overlap.
• Embedded index coding: When $N=K$ and $M_n = R(n)$, hyperminrank recovers known optimal rates.

In all cases, minimizing $\operatorname{hyper\text{-}minrank}(\mathcal{G})$ offers a unified metric for both the design of message placement across senders and the delivery of scalar linear transmissions, supporting efficient, theoretically-optimal coding strategies in practical multi-terminal scenarios (Khalesi et al., 15 Dec 2025).